DIFFERENTIAL ITEM FUNCTIONING AND

COGNITIVE ASSESSMENTUSING IRT-BASED METHODS

Jeanne Teresi, Ed.D., Ph.D.

Katja Ocepek-Welikson, M.Phil.

PART I: OVERVIEW

Jeanne Teresi, Ed.D., Ph.D.

• A recent report on national healthcare disparities (DHHS, Agency for Healthcare Research and Quality, National Healthcare Disparities Report, 2003) concluded that:

• “Disparities in the health care system are pervasive”

• “Racial, ethnic and socioeconomic disparities are national problems that affect health care…”

• Differential item functioning analyses are important in health disparities research

USES OF DIF ANALYSES:

• EVALUATE EXISTING MEASURES

• DEVELOP NEW MEASURES THAT ARE AIMED TO BE:– Culture Fair– Gender Equivalent– Age invariant

DIF METHODS

There are numerous review articles and books related to DIF. A few are:

• Camilli and Shepard, 1994

• Holland and Wainer; 1993

• Millsap and Everson, 1993

• Potenza and Dorans, 1995

• Thissen, Steinberg and Wainer, 1993

DEFINITIONS

• DIF INVOLVES THREE FACTORS:

– Response to an item

– Conditioning/matching cognitive status variable

– Background (grouping) variable(s)

• DIF can be defined as conditional probabilities or conditional expected item scores that vary across groups.

A randomly-selected person of average cognitive function interviewed in Spanish should have the same chance of responding in the unimpaired direction to a cognitive status item as would a randomly selected person of average function interviewed in English

CONTROLLING FOR LEVEL OF COGNITIVE STATUS, IS RESPONSE TO AN ITEM

RELATED TO GROUP MEMBERSHIP?

EXAMPLE

• Contingency table that examines the cross-tabulation of item response by group membership for every level (or grouped levels) of the attribute estimate

Two by two contingency table for item ‘Does not State Correct State’ by language groups, conditioning on the MMSE summary score (score levels 8 to 12)

Item Score

GroupNo Error

(0)Incorrect

(1)Total

Focal (English interview)221

(90.2%)24

(9.8%)245

(100%)

Reference group (Spanish interview)

113 (62.1%)

69 (37.9%)

182 (100%)

Total334

(78.2%)93

(21.8%)427

UNIFORM DIF DEFINITIONS

• DIF is in the same direction across the entire spectrum of disability (item response curves for two groups do not cross)

• DIF involves the location (b) parameters

• DIF is a significant main (group) effect in regression analyses predicting item response

MMSE 21 Item Scale, Item 6 - Does Not State Correct State Boundary Response Functions (for k = categories 0, 1)

Comparing Two Language Groups

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Cognitive Dysfunction (Theta)

Pro

ba

bili

ty o

f R

esp

on

din

g I

nco

rre

ctly

(P

i)

English, Category 1:a=2.34,b=.97

Spanish, Category 1:a=1.55,b=.12

• The probability of a randomly selected Spanish speaking person of mild cognitive dysfunction (theta = 0) responding incorrectly to the item “Does not State Correct State” is higher (.45) than for a randomly selected English speaking person (.09) at the same cognitive dysfunction level. (Given equal cognitive dysfunction, Spanish speaking respondents are more likely than English speaking respondents to make an error.)

NON-UNIFORM DIF

• An item favors one group at certain disability levels, and other groups at other levels (or the probability of item endorsement is higher for group 1 at lower ability and higher for group 2 at higher ability)

• DIF involves the discrimination (a) parameters• DIF is a significant group by ability interaction in

regressions predicting item response • DIF is assessed by examination of nested models

comparing differences in log-likelihoods

Physical Functioning Item Set, Item 22 - Walking One BlockPlot of Boundary Response Functions (for k = categories 0, 1, 2)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Ability (Theta)

Pro

ba

bili

ty o

f R

esp

on

din

g in

Ca

teg

ory

k o

r H

igh

er

(Pi)

Whites, Category 1:a=3.53,b= -.90

Category 2:a=3.53,b= -.07

Afr-Amer, Category 1:a=2.64,b=-1.19

Category 2:a=2.64,b= -.01

MMSE 21 Item Scale, Item 19 - Errors Following Instructions With Paper Boundary Response Functions (for k = categories 0, 1, 2, 3)

Comparing Two Language Groups

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Cognitive Dysfunction (Theta)

Pro

ba

bili

ty o

f R

esp

on

din

g in

Ca

teg

ory

k o

r H

igh

er

(Pi)

English, Category 1:a=2.20,b= -.14 Category 2:a=2.20,b= .80 Category 3:a=2.20,b=1.57Spanish, Category 1:a=2.20,b= .26 Category 2:a=2.20,b=1.18 Category 3:a=2.20,b=1.70

MAGNITUDE

• Magnitude of DIF

Item level characteristic, e.g.,

odds ratio,

area statistic,

beta coefficient or R square increment,

expected item scores

Physical Functioning Item Set, Item 22 - Walking One BlockExpected Item Score by Race Groups

0.0

0.5

1.0

1.5

2.0

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Ability (Theta)

Exp

ecte

d It

em

Sco

re

Whites

African-American

IMPACTImpact in the context of cognitive measures:• Differences in the cognitive status distributions

and summary statistics between or among studied groups

• Group differences in the total (test) response function

• Group differences in relationship of demographic variables to cognitive status variables with and without adjustment for DIF

MMSE 21Item Scale Test Response Functions

Comparing Two Language Groups

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Cognitive Dysfunction (Theta)

Expe

cted

Tot

al S

cale

Sco

re

English

Spanish

IRT-BASED METHODS

• Likelihood ratio test based on IRT (Thissen, 1991, 2001)– Based on examination of differences in

fit between compact and augmented models that include additional free parameters representing non-uniform and uniform DIF

– Latent conditioning variable

SOME ADVANTAGES OF IRTLR

• Well-developed theoretical models• Can examine uniform and non-uniform DIF• No equating required because of simultaneous

estimation of group parameters• Can model missing data• Simulations show superior performance (in terms

of power, particularly with small sample sizes) in comparison with non-parametric methods (Bolt, 2002)

• Model must fit the data; misfit results in Type I error Inflation (Bolt, 2002)• Requires categorical group variable• Assumptions must be met• Magnitude measures not as well-integrated• No formal magnitude summary measure or guidelines

POSSIBLE DISADVANTAGESOF IRTLR

AREA AND DFIT METHODSArea and DFIT methods based on IRT model with latent conditioning variable (Raju and colleagues, 1995; Flowers and colleagues, 1999)

Non-compensatory DIF (NCDIF) indices:average squared differences in item “true” or expected raw scores for individuals as members of the focal group and as members of the reference group (expected score is the sum of the (weighted) probabilities of category endorsement, conditional on disability).

Differential test functioning (DTF) :based on the compensatory DIF (CDIF) index and reflects group differences summed across items

SOME ADVANTAGES OF DFIT• Can detect both uniform and non-uniform

DIF, and shares the advantages of IRT models upon which it is based

• Magnitude measures used for DIF detection

• Impact of item DIF on the total score is examined

• One simulation study (in comparison with IRTLR) showed favorable performance in terms of false DIF detection (Bolt, 2002)

SOME DISADVANTAGES OF DFIT

• Requires parameter equating• Many programs required for DIF testing• Model misfit will result in false DIF detection • χ2 statistical tests affected by sample size, and identification of optimal cut-points for DIF detection requires further simulation

DIFFERENCES AMONG DIF METHODS CAN BE CHARACTERIZED

ACCORDING TO WHETHER THEY:

• Are parametric or non-parametric• Are based on latent or observed variables• Treat the disability dimension as continuous • Can model multiple traits • Can detect both uniform and non-uniform DIF• Can examine polytomous responses• Can include covariates in the model• Must use a categorical studied (group variable)

CONCLUSIONS

• DIF cancellation at the aggregate level may still have an impact on an individual

• DIF assessment of measures remains a critical component of health disparities research, and of efforts to achieve cultural equivalence in an increasingly, culturally diverse society

PART II: STEPS IN IRTLRDIF ANALYSIS

Katja Ocepek-Welikson, M.Phil.

IRTLRDIF ANALYSIS

The underlying procedure of IRTLRDIF is a series of comparisons of compact and augmented models. Likelihood ratio tests are used for comparison resulting in goodness of fit statistic G2 distributed as a χ2

STEP 1: NO ANCHOR ITEMS DEFINED

STEP 1a:The first comparison is between a model with all parameters constrained to be equal for the two groups, including the studied item, with a model with separate estimation of all parameters for the studied item.

IRTLRDIF is designed using stringent criteria for DIF detection, so that if any model comparison results in a χ2 value greater than 3.84 (d.f.= 1), indicating that at least one parameter differs between the two groups at the .05 level, the item is assumed to have DIF.

STEP 1b:If there is any DIF, further model comparisons are performed

STEP 1c:Two-parameter models, test of DIF in the ‘a’ parameter: the model with all parameters constrained is compared to a model in which the ‘a’ parameter (slope or discrimination) is constrained to be equal and the ‘b’ parameter (difficulty or threshold) is estimated freely

STEP 1d:The same concepts are followed with respect to the ‘b’ parameters test of DIF. The ‘a’ parameters are constrained equal and the ‘b’ parameters are free to be estimated as different.

The G2 for this last model is derived by subtraction of the G2 for evaluation of the ‘a’ parameters from the overall G2 value evaluating any difference (G2 all equal - G2 a's equal).

STEP 2: ANCHOR ITEM SET

• For all models, all items are constrained to be equal within the anchor set

• Anchor items are defined as those with the G2 cutoff value of 3.84 or less for the overall test of all parameters equal versus all parameters free for the studied item (for a dichotomous item under the 2p model the d.f. = 2)

This may result in the selection of a very small anchor set for some comparisons. Therefore, these criteria may be relaxed somewhat, and the results of the individual parameter estimates examined rather than the overall result. If significant DIF is observed for the a's or b's using appropriate degrees of freedom, then the item will be excluded from the anchor set.

ANCHOR ITEM SET, cont.

FINAL ANCHOR ITEM SET

Even if anchor items were identified prior to the analyses using IRTLRDIF, additional items with DIF may be identified. All of the items in the anchor test are again evaluated, following the procedures described in step 1, in order to exclude any additional items with DIF, and to finalize the anchor set.

STEP 3: FINAL TESTS FOR DIF

After the anchor item set is defined, all of the remaining (non-anchor) items are evaluated for DIF against this anchor set. Some items that have been identified as having DIF in earlier stages of the analyses, can convert to non-DIF with the use of a purified anchor set.

(It is noted that the studied item is modeled along with the anchor items, so that parameter estimates are based on the anchor item set with inclusion of the studied item.)

STEP4: ADJUSTMENT FOR MULTIPLE COMPARISONS

Items with values of G2 indicative of DIF in this last stage are subject to adjustment or p values for multiple comparisons used in order to reduce over-identification of items with DIF. Bonferroni, Benjamini-Hochberg or other comparable method to control for false discovery can be used.

STEP 5: MULTILOG RUN TO OBTAIN FINAL PARAMETER

ESTIMATES• In order to obtain the final item parameter

estimates, an additional MULTILOG run has to be performed

• Parameters are estimated simultaneously for two groups

• Parameters for anchor items are set to be estimated equal for two groups

• Parameters for items with DIF are estimated separately (if only ‘b’ parameters show DIF, ‘a’s are set as equal)

SUMMARY OF STEPS IN DFIT ANALYSIS

1. Perform an assessment of dimensionality2. Perform IRT analyses to obtain

parameters and disability estimates; perform analyses separately for each group (both PARSCALE and MULTILOG can be used)

3. Equate the parameters (Baker’s EQUATE program was used in this step)

DFIT STEP, cont.

4. Identify DIF using DFIT (DFIT5P was used)

5. Identify anchor items that are relatively DIF-free, using NCDIF cutoffs rather than χ2 significance tests that are available

6. Purify the equating constants by re-equating

7. Perform DFIT again

DFIT STEP, cont.

8. Examine the NCDIF cutoffs to determine items with DIF

9. Examine CDIF and DTF to determine if values exceed the cutoff, indicating differential test (scale) functioning

10. If DTF > the cutoff, examine the removal index to identify items that might be removed

DFIT STEP, cont.

11. Calculate expected item scores; sum the expected item scores to obtain an expected test (scale) score for each group, separately

12. Plot the expected scale scores against theta (disability) for each group